We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools; mean service times depend both on the customer class and server pool. [Such systems model, e.g., large call/contact centers with agents having different sets of "skills".] It is assumed that the allowed activities (routing choices) form a tree in the bipartite graph with vertices being customer classes and server pools). We study a natural load-balancing routing/scheduling rule, Longest-queue freest-server (LQFS-LB), in the many-server asymptotic regime: the exogenous arrival rates of the customer classes, as well as the number of agents in each pool, grow to infinity in proportion to some scaling parameter r. Equilibrium point of the system under LQBS-LB is the desired operating point, with server pool loads minimized and perfectly balanced.
Main results: (a) We show that, quite surprisingly (given the tree assumption), for certain parameter ranges, the fluid limit of the system may be unstable in the vicinity of the equilibrium point; such instability may occur if the activity graph is not "too small". (b) Using (a), we demonstrate that the sequence of stationary distributions of diffusion-scaled processes (measuring O(sqrt{r}) deviations from the equilibrium point) may be non-tight, and in fact may escape to infinity. (c) In the special case when mean service times depend on the server pool only,
we show that the sequence of stationary distributions of diffusion-scaled processes is tight, and therefore the limit of stationary distributions is the stationary distribution of the limiting diffusion process.
Joint work with Elena Yudovina (Cambridge)